Abstract The starting point of
our approach to the number systems is the selection of basic
operative properties of the system $N_0$ of natural numbers with~
$0$. This set of properties has proved itself to be sufficient for
extension of this system to the systems of integers $Z$,
positive rational numbers $Q_+\cup\{0\}$ and rational numbers
$Q$ and for the formation of basic operative properties of these
extended systems.
\par
In all these cases of number systems, the corresponding set of
numbers with basic operative properties is an example of a
concrete algebraic structure. These structures can be viewed
abstractly as the structure $(S, +, \cdot, <)$, where $S$ is a
nonempty set, ``$+$'' and ``$\cdot$'' are two binary operations
on $S$ and ``$<$'' is the order relation on $S$, which satisfy the
postulated conditions that are formed according to the basic
operative properties of these systems. When matched up with
$N_0$, $Z$, $Q_+\cup\{0\}$ and $Q$, the structure $(S,+,\cdot,<)$
is called ordered semifield, ordered semifield with
additive inverse, ordered semifield with multiplicative inverse
and ordered field, respectively. Then, these number systems are
characterized as being the smallest semifield with which they fit
together. Proofs of these facts require deduction of some
properties of all mentioned types of this abstract structure upon
which they will be clearly relied. Hence, the main aim of this
paper is this deduction and some improvements of proofs contained
in the paper whose reconsideration is this note.
